Find the best tutors and institutes for Class 12 Tuition
Q23:
Find all points of discontinuity of $f$, where $f(x) = \begin{cases} \frac{\sin x}{x}, & \text{if } x < 0 \\ x + 1, & \text{if } x \ge 0 \end{cases}$
Solution :
Step 1: Analyzing the Domain of Continuity
The function $f(x)$ is defined piecewise over the set of real numbers $\mathbb{R}$. We must test for continuity at the point of interest, $x = 0$, where the definition of the function transitions, and evaluate the function behavior in the open intervals $(-\infty, 0)$ and $(0, \infty)$.
Step 2: Analysis at $x < 0$
For $x \in (-\infty, 0)$, $f(x) = \frac{\sin x}{x}$.
Since $g(x) = \sin x$ and $h(x) = x$ are continuous functions on their entire domains, their quotient $\frac{g(x)}{h(x)}$ is continuous provided the denominator $h(x) \neq 0$ [Per the Quotient Rule for Continuous Functions]. Since $x \neq 0$ in this interval, $f(x)$ is continuous for all $x < 0$.
Step 3: Analysis at $x > 0$
For $x \in (0, \infty)$, $f(x) = x + 1$.
This is a polynomial function of degree 1. Polynomials are continuous for all $x \in \mathbb{R}$ [Per the Polynomial Continuity Theorem]. Thus, $f(x)$ is continuous for all $x > 0$.
Step 4: Analysis at the Critical Point $x = 0$
To determine if $f(x)$ is continuous at $x = 0$, we must satisfy the condition $\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0)$.
- Left-Hand Limit (LHL): $\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} \frac{\sin x}{x}$. Applying the standard trigonometric limit, $\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$ [By L'Hôpital's Rule or Small Angle Approximation Taylor Series], the LHL = 1.
- Right-Hand Limit (RHL): $\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (x + 1) = 0 + 1 = 1$.
- Function Value at $x = 0$: $f(0) = 0 + 1 = 1$ (using the definition for $x \ge 0$).
Since $\text{LHL} = \text{RHL} = f(0) = 1$, the function $f(x)$ is continuous at $x = 0$.
Step 5: Visual Representation
Summary Analysis
| Interval | Continuity Status |
|---|---|
| $x < 0$ | Continuous |
| $x = 0$ | Continuous |
| $x > 0$ | Continuous |
Final Solution: The function $f(x)$ is continuous for all $x \in \mathbb{R}$. Therefore, there are no points of discontinuity.
More Questions from Class 12 Mathematics Continuity and Differentiability EXERCISE 5.1
- Q1: Prove that the function $f(x) = 5x - 3$ is continuous at $x = 0$, at $x = -3$ and at $x = 5$.
- Q2: Examine the continuity of the function $f(x) = 2x^2 - 1$ at $x = 3$.
- Q3(a): Examine the following functions for continuity. (a) $f(x) = x - 5$
- Q3(b): Examine the following functions for continuity. (b) $f(x) = \frac{1}{x-5}$, $x \neq 5$
- Q3(c): Examine the following functions for continuity. (c) $f(x) = \frac{x^2 - 25}{x+5}$, $x \neq -5$
- Q3(d): Examine the following functions for continuity. (d) $f(x) = |x - 5|$
- Q4: Prove that the function $f(x) = x^n$ is continuous at $x = n$, where $n$ is a positive integer.
- Q5: Is the function $f$ defined by $f(x) = \begin{cases} x, & \text{if } x \le 1 \\ 5, & \text{if } x > 1 \end{cases}$ continuous at $x = 0$? At $x = 1$? At $x = 2$?
- Q6: Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} 2x + 3, & \text{if } x \le 2 \\ 2x - 3, & \text{if } x > 2 \end{cases}$
- Q7: Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} |x| + 3, & \text{if } x \le -3 \\ -2x, & \text{if } -3 < x < 3 \\ 6x + 2, & \text{if } x \ge 3 \end{cases}$
- Q8: Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} \frac{|x|}{x}, & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases}$
- Q9: Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} \frac{x}{|x|}, & \text{if } x < 0 \\ -1, & \text{if } x \ge 0 \end{cases}$
- Q10: Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} x + 1, & \text{if } x \ge 1 \\ x^2 + 1, & \text{if } x < 1 \end{cases}$
- Q11: Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} x^3 - 3, & \text{if } x \le 2 \\ x^2 + 1, & \text{if } x > 2 \end{cases}$
- Q12: Find all points of discontinuity of $f$, where $f$ is defined by $f(x) = \begin{cases} x^{10} - 1, & \text{if } x \le 1 \\ x^2, & \text{if } x > 1 \end{cases}$
- Q13: Is the function defined by $f(x) = \begin{cases} x + 5, & \text{if } x \le 1 \\ x - 5, & \text{if } x > 1 \end{cases}$ a continuous function?
- Q14: Discuss the continuity of the function $f$, where $f$ is defined by $f(x) = \begin{cases} 3, & \text{if } 0 \le x \le 1 \\ 4, & \text{if } 1 < x < 3 \\ 5, & \text{if } 3 \le x \le 10 \end{cases}$
- Q15: Discuss the continuity of the function $f$, where $f$ is defined by $f(x) = \begin{cases} 2x, & \text{if } x < 0 \\ 0, & \text{if } 0 \le x \le 1 \\ 4x, & \text{if } x > 1 \end{cases}$
- Q16: Discuss the continuity of the function $f$, where $f$ is defined by $f(x) = \begin{cases} -2, & \text{if } x \le -1 \\ 2x, & \text{if } -1 < x \le 1 \\ 2, & \text{if } x > 1 \end{cases}$
- Q17: Find the relationship between $a$ and $b$ so that the function $f$ defined by $f(x) = \begin{cases} ax + 1, & \text{if } x \le 3 \\ bx + 3, & \text{if } x > 3 \end{cases}$ is continuous at $x = 3$.
- Q18: For what value of $\lambda$ is the function defined by $f(x) = \begin{cases} \lambda(x^2 - 2x), & \text{if } x \le 0 \\ 4x + 1, & \text{if } x > 0 \end{cases}$ continuous at $x = 0$? What about continuity at $x = 1$?
- Q19: Show that the function defined by $g(x) = x - [x]$ is discontinuous at all integral points. Here $[x]$ denotes the greatest integer less than or equal to $x$.
- Q20: Is the function defined by $f(x) = x^2 - \sin x + 5$ continuous at $x = \pi$?
- Q21(a): Discuss the continuity of the following functions: (a) $f(x) = \sin x + \cos x$
- Q21(b): Discuss the continuity of the following functions: (b) $f(x) = \sin x - \cos x$
- Q21(c): Discuss the continuity of the following functions: (c) $f(x) = \sin x \cdot \cos x$
- Q22: Discuss the continuity of the cosine, cosecant, secant and cotangent functions.
- Q24: Determine if $f$ defined by $f(x) = \begin{cases} x^2 \sin \frac{1}{x}, & \text{if } x \neq 0 \\ 0, & \text{if } x = 0 \end{cases}$ is a continuous function?
- Q25: Examine the continuity of $f$, where $f$ is defined by $f(x) = \begin{cases} \sin x - \cos x, & \text{if } x \neq 0 \\ -1, & \text{if } x = 0 \end{cases}$
- Q26: Find the values of $k$ so that the function $f$ is continuous at the indicated point in Exercises 26 to 29. $f(x) = \begin{cases} \frac{k \cos x}{\pi - 2x}, & \text{if } x \neq \frac{\pi}{2} \\ 3, & \text{if } x = \frac{\pi}{2} \end{cases}$ at $x = \frac{\pi}{2}$
- Q27: Find the values of $k$ so that the function $f$ is continuous at the indicated point in Exercises 26 to 29. $f(x) = \begin{cases} kx^2, & \text{if } x \le 2 \\ 3, & \text{if } x > 2 \end{cases}$ at $x = 2$
- Q28: Find the values of $k$ so that the function $f$ is continuous at the indicated point in Exercises 26 to 29. $f(x) = \begin{cases} kx + 1, & \text{if } x \le \pi \\ \cos x, & \text{if } x > \pi \end{cases}$ at $x = \pi$
- Q29: Find the values of $k$ so that the function $f$ is continuous at the indicated point in Exercises 26 to 29. $f(x) = \begin{cases} kx + 1, & \text{if } x \le 5 \\ 3x - 5, & \text{if } x > 5 \end{cases}$ at $x = 5$
- Q30: Find the values of $a$ and $b$ such that the function defined by $f(x) = \begin{cases} 5, & \text{if } x \le 2 \\ ax + b, & \text{if } 2 < x < 10 \\ 21, & \text{if } x \ge 10 \end{cases}$ is a continuous function.
- Q31: Show that the function defined by $f(x) = \cos(x^2)$ is a continuous function.
- Q32: Show that the function defined by $f(x) = |\cos x|$ is a continuous function.
- Q33: Examine that $\sin|x|$ is a continuous function.
- Q34: Find all the points of discontinuity of $f$ defined by $f(x) = |x| - |x + 1|$.
CBSE Solutions for Class 12 Mathematics Continuity and Differentiability
- Continuity and Differentiability Class 12 EXERCISE 5.2
- Continuity and Differentiability Class 12 EXERCISE 5.3
- Continuity and Differentiability Class 12 EXERCISE 5.4
- Continuity and Differentiability Class 12 EXERCISE 5.5
- Continuity and Differentiability Class 12 EXERCISE 5.6
- Continuity and Differentiability Class 12 EXERCISE 5.7
Chapters in CBSE - Class 12 Mathematics
Top Tutors who teach Continuity and Differentiability
Top Tutor
Ambala, Ambala
10 yrs of
Exp
4 students
277 hrs of classes completed
185
per hour
Online Classes
I am a History and law graduate, specializing in coaching students appearing for their board exams. From clearing concepts to teaching how to efficiently write answers, I do it all. For Class 12, I focus on: * Clear understanding of concepts, thinkers & case studies * Answer writing practice as per board requirements * Important questions, PYQs & expected topics * Smart revision techniques to improve scores My classes are one-on-one and tailored to your pace so that you feel confident before exams.
I had a great experience with Kanika Ma’am. She taught me political science for my boards. She is a very knowledgeable and kind teacher. Must recommend.
Top Tutor
Naskar Bagan, Kolkata
1 yrs of
Exp
3 students
206 hrs of classes completed
400
per hour
Online Classes
Student's Home
🚀 High-Impact Computer Science & IP Tutoring for CBSE& CISCE Class 10, 11 & 12 Hi! I’m a B.C.A. graduate and experienced online educator dedicated to helping students build genuine logical thinking and ace their board examinations. Having scored 94/100 in my own Class 12 Computer Science boards using highly efficient study strategies, I know exactly how to simplify the syllabus for maximum understanding and top marks. I bridge the gap between memorizing syntax and actually understanding how to code. 🌟 Why Choose My Classes? 1. Targeted Expertise: Specialized coaching for CBSE & CISCE Class 10 ,11 & 12 Computer Science, Informatics Practices (IP), and Information Technology (including Java, Python programming and database management). 2. Proven Resources: Standard reference alignment, custom study notes, topic-wise assignments, and rigorous mock tests. 3. Interactive Learning: 100% online sessions utilizing dynamic PPTs and live whiteboards—no boring lectures, just hands-on, practical coding. 4. Track Record: Successfully coached students from top schools to become confident, independent problem solvers. 5. My Approach: I don't just teach you how to write code; I teach you how to think like a programmer so you can approach any exam question with confidence. Let’s turn coding anxiety into board exam success.
Great thought Shares real-life experiences Great mentor Includes most of the important topics in the demo session.
Top Tutor
Sector 54 Suncity, Gurgaon
8 yrs of
Exp
1185 hrs of classes completed
800
per hour
Online Classes
Tutor's home
I have a varied experience of over 16 years and have enjoyed all roles as a writer, editor, banker, business analyst, teacher and marketer. I love a challenge, and during the course of my career, I took up English Language classes for working professionals. My classes are learner-centric and interactive, packed with activities, games and exercises that draw learners out of their shells and infuse them with confidence. Students are encouraged to think, raise questions, absorb new vocabulary and finally adopt the language as their own. My ability to connect and engage with students from all sections make my classes friendly, approachable and non-intimidating. My varied work experience across sectors further adds to my understanding of job and situation requirements.
Top Tutor
Horamavu, Bangalore
4 yrs of
Exp
7 students
176 hrs of classes completed
400
per hour
Online Classes
Tutor's home
As a technology and data professional, my approach to teaching Class 12 Mathematics bridges the gap between abstract textbook formulas and clear, logical problem-solving execution. I specialize in training students for both the CBSE Board and State Board examinations, ensuring a highly structured approach to scoring maximum marks. My Class 12 Mathematics Instructional Blueprint covers: • Calculus Mastery: Simplifying high-weightage topics like Continuity & Differentiability, Integrals, and Applications of Derivatives (AoD) through step-by-step logical breakdowns. • Vectors & 3D Geometry: Using visual and coordinate tools to make highly abstract geometric concepts easy to grasp and reproduce in exams. • Algebra & Probability: Covering Matrices, Determinants, and Relations & Functions with complete conceptual clarity. Exam-Targeted Strategy: • Board-Specific Focus: Comprehensive coverage of NCERT for CBSE and prescribed text-books for State Boards, focusing on question-solving patterns from the past 10 years of board papers. • Marks-Scoring Techniques: Teaching students exactly how to present their steps in the answer sheet to claim full step-marking. • Regular Assessments: Weekly topic-wise tests and time-bound mock exams to track speed and accuracy. My teaching style is patient, structured, and collaborative. I ensure that concepts are thoroughly understood before moving to advanced problems, creating a supportive online environment where students feel completely comfortable clearing their doubts. Let's build a strong mathematical foundation and secure top scores in the upcoming Board Exams! Contact me to book a demo session.
I had a few demo sessions with her and i found her really approachable, patient and fun to learn with. She made sure that i understood, was patient and understanding enough. I really enjoyed learning from her.
Super Tutor
Parsudih, Jamshedpur
5 yrs of
Exp
30 students
2586 hrs of classes completed
600
per hour
Online Classes
Tutor's home
I teach each student with utmost care. Students love my classes because it is very interactive and informative. Class 12 is very important not only because of the board exam but also because it is the staircase towards professional career. So, study content, right guidance to achieve success is the need of the hour. I am happy to help my students at each difficult challenges to bring success.
Simply he is doing very well, and my child is also learning the things with great enthusiasm.Debojit sarkar ji is simply phenomenal
Other Subjects in CBSE - Class 12
Download free CBSE - Class 12 Mathematics Continuity and Differentiability EXERCISE 5.1 worksheets
Download Now
